1. Pushdown Automata PDAs
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2. Pushdown Automaton -- PDA
Input String
Stack
States
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3. Formal Definition Non-Deterministic Pushdown Automaton NPDA Final
states
States
Input
alphabet
Stack
start
symbol
Transition
function
Initial
state
Stack
alphabet
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4. Initial Stack Symbol Stack Stack stack head top bottom special symbol
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5. The States
Pop
symbol
Input
symbol
Push
symbol
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6. input
stack
top
Replace
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7. input
stack
top
Push
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8. input
stack
top
Pop
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9. input
stack
top
No Change
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10. A Possible Transition
input
stack
empty
Pop
top
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11. The automaton Halts in state and Rejects the input string
A Bad Transition
input
Empty stack
HALT
The automaton Halts in state
and Rejects the input string
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12. The automaton Halts in state and Rejects the input string
A Bad Transition
input
Empty stack
HALT
The automaton Halts in state
and Rejects the input string
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13. No transition is allowed to be followed When the stack is empty
Empty stack
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14. A Good Transition
input
stack
Pop
top
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15. Non-Determinism These are allowed transitions in a
Non-deterministic PDA (NPDA)
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16. NPDA: Non-Deterministic PDA
Example:
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17. Execution Example:
Time 0
Input
Stack
current
state
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18. Time 1
Input
Stack
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19. Time 2
Input
Stack
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20. Time 3
Input
Stack
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21. Time 4
Input
Stack
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22. Time 5
Input
Stack
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23. Time 6
Input
Stack
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24. Time 7
Input
Stack
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25. Time 8
Input
Stack
accept
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26. A string is accepted if there is a computation such that:
All the input is consumed
AND
The last state is a final state
At the end of the computation,
we do not care about the stack contents
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27. is accepted by the NPDA:
The input string
is accepted by the NPDA:
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28. is the language accepted by the NPDA:
In general,
is the language accepted by the NPDA:
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29. Another NPDA example
NPDA
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30. Execution Example:
Time 0
Input
Stack
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31. Time 1
Input
Stack
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32. Time 2
Input
Stack
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33. Time 3
Input
Guess the middle
of string
Stack
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34. Time 4
Input
Stack
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35. Time 5
Input
Stack
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36. Time 6
Input
Stack
accept
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37. Rejection Example:
Time 0
Input
Stack
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38. Time 1
Input
Stack
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39. Time 2
Input
Stack
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40. Time 3
Input
Guess the middle
of string
Stack
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41. Time 4
Input
Stack
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42. There is no possible transition. Input
Time 5
There is no possible transition.
Input
Input is not consumed
Stack
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43. Another computation on same string:
Input
Time 0
Stack
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44. Time 1
Input
Stack
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45. Time 2
Input
Stack
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46. Time 3
Input
Stack
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47. Time 4
Input
Stack
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48. Time 5
Input
No final state
is reached
Stack
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49. There is no computation that accepts string
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50. A string is rejected if there is no computation such that:
All the input is consumed
AND
The last state is a final state
At the end of the computation,
we do not care about the stack contents
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51. In other words, a string is rejected
if in every computation with this string:
The input cannot be consumed OR
The input is consumed and the last state is not a final state
The stack head moves below the bottom of the stack
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52. Another NPDA example
NPDA
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53. Execution Example:
Time 0
Input
Stack
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54. Time 1
Input
Stack
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55. Time 2
Input
Stack
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56. Time 3
Input
Stack
accept
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57. Rejection example:
Time 0
Input
Stack
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58. Time 1
Input
Stack
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59. Time 2
Input
Stack
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60. Time 3
Input
Stack
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61. Time 4
Input
Stack
Halt and Reject
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62. Pushing Strings
Pop
symbol
Input
symbol
Push
string
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63. Example:
input
pushed
string
stack
top
Push
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64. Another NPDA example
NPDA
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65. Execution Example:
Time 0
Input
Stack
current
state
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66. Time 1
Input
Stack
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67. Time 3
Input
Stack
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68. Time 4
Input
Stack
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69. Time 5
Input
Stack
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70. Time 6
Input
Stack
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71. Time 7
Input
Stack
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72. Time 8
Input
Stack
accept
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73. Formalities for NPDAs
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74. Transition function:
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75. Transition function:
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76. Formal Definition Non-Deterministic Pushdown Automaton NPDA Final
states
States
Input
alphabet
Stack
start
symbol
Transition
function
Initial
state
Stack
alphabet
Costas Busch - RPI